
theorem Th21:
  for L be non empty multMagma for B be non empty AlgebraStr over
  L for A be non empty Subset of B st A is opers_closed holds the carrier of
  GenAlg A = A
proof
  let L be non empty multMagma;
  let B be non empty AlgebraStr over L;
  let A be non empty Subset of B;
  assume A is opers_closed;
  then ex C be strict Subalgebra of B st the carrier of C = A by Th19;
  then
A1: the carrier of GenAlg A c= A by Def5;
  A c= the carrier of GenAlg A by Def5;
  hence thesis by A1,XBOOLE_0:def 10;
end;
